MCH eye bone animation plays in Blender, but not in Unity
OK, I've figured it out. I had parented the eye meshes to the 2 MCH bones, but I hadn't added any vertex weights. I just had to set all vertices in the eye spheres to a vertex group with a name that matched the correct eye bone at weight '1' and it's now all sorted in Unity. I'm glad I checked Unity once I made the change because the eye animation is actually now messed up in Blender - The eyes at certain points swing about to odd positions. But it's working in Unity so I'm not going to complain.

EDIT:

I've now figured out how to get it looking right both in Blender and Unity. Turns out the eye meshes should be parented to the armature just like everything else. I just previously hadn't used the right bone for the vertex weighting. Using the MCH_eye.L and .R bones as the vertex groups did the trick. Looks right in Blender and Unity. A relief when things make sense properly.